Problem 25 - College Algebra

Question

For the following exercises, rewrite each expression as an equivalent ratio of logs using the indicated base. $$ \log _{7}(15) \text { to base } e $$

Step-by-Step Solution

Answer

\(\log_7(15) = \frac{\ln(15)}{\ln(7)}\)

Step 1: Understand the Change of Base Formula

The change of base formula for logarithms allows us to convert a logarithm of one base to another base. The formula is given by:\[\log_b(a) = \frac{\log_k(a)}{\log_k(b)}\]where \(a\) is the number, \(b\) is the original base, and \(k\) is the new base we want. We can apply this formula to the given problem.

Step 2: Identify the Values

In the exercise, we have \(\log_7(15)\) and we want to rewrite it to base \(e\). Here, \(a = 15\), \(b = 7\), and \(k = e\). These values need to be substituted into the change of base formula.

Step 3: Substitute Values into the Change of Base Formula

Using the values identified, substitute into the change of base formula:\[\log_7(15) = \frac{\log_e(15)}{\log_e(7)}\]This means that we express the logarithm to base 7 in terms of natural logarithms (i.e., base \(e\)).

Step 4: Simplify the Expression

The expression \(\log_7(15)\) in terms of base \(e\) is now given by the fraction of logs:\[\frac{\ln(15)}{\ln(7)}\]where \(\ln\) denotes the natural logarithm, which is the logarithm to base \(e\).

Key Concepts

Understanding Logarithms

Logarithms are mathematical expressions that represent the power to which a number, called the base, must be raised to produce a given number. To put it simply, if you have a number represented as a power, logarithms help you find the exponent. Logarithms can be used in various mathematical contexts, and they are denoted as \ \( \log_b(a) \ \), where \ \( b \ \) is the base and \ \( a \ \) is the number for which we are finding the log. Understanding the foundational concept of logarithms involves:

• The base, often denoted as \ \( b \ \), which is the number that gets raised to a power.
• The number \ \( a \ \), which is the result you are starting with.
• The logarithm itself, which tells you the power or exponent. For example, in \ \( \log_2(8) = 3 \ \), 2 is the base and 8 is the number; thus, 2 must be raised to the power of 3 to equal 8.

Logarithms can be calculated in different bases, commonly 10 (common logarithms) and the natural number \ \( e \ \) (natural logarithms).

Exploring Natural Logarithms

Natural logarithms are a special type of logarithm where the base is the constant \ \( e \ \), approximately equal to 2.71828. This base is unique and arises naturally in calculus and natural growth processes. The natural logarithm of a number \ \( a \ \) is represented as \ \( \ln(a) \ \). Natural logarithms have several properties useful in mathematical calculations:

• They are the inverse functions of exponential functions. This means \ \( \ln(e^x) = x \ \) and \ \( e^{\ln(x)} = x \ \).
• The derivative of \ \( \ln(x) \ \) with respect to \ \( x \ \) is \ \( \frac{1}{x} \ \).
• They simplify calculations involving growth rates, such as interest compounded continuously.

The natural logarithm appears frequently in advanced mathematics and science, especially when dealing with growth phenomena, physical constants, and complex systems.

The Process of Base Conversion

Converting between different bases of logarithms is an essential skill in mathematics. This process is often facilitated by the Change of Base Formula, which enables the conversion of a logarithm from one base to another. The formula is:\[ \log_b(a) = \frac{\log_k(a)}{\log_k(b)} \] where:

• \ \( b \ \) is the original base of the logarithm.
• \ \( a \ \) is the number you want to find the log of.
• \ \( k \ \) is the new base you want to convert to.

In practical terms, this means that any logarithm can be rewritten in terms of logs of a different base using this ratio format. For instance, as illustrated in the original exercise, the logarithm \ \( \log_7(15) \ \) is converted to a natural logarithm (base \ \( e \ \)) as follows: \[ \log_7(15) = \frac{\ln(15)}{\ln(7)} \] This conversion makes calculations with non-standard bases manageable without the need for specialized calculators. It is especially useful in scientific and engineering applications where the natural logarithm base \ \( e \ \) is prevalent.